Virtual Distillation for Quantum Error Mitigation

1. A method for determining an error-mitigated expectation value of a target observable with respect to a noisy quantum state, the method performed by a system comprising a classical processor and a quantum computer, the method comprising: obtaining, by the quantum computer, multiple copies of the noisy quantum state, wherein the noisy quantum state comprises stochastic errors resulting from noise on the quantum computer; performing, by the quantum computer, measurements on tensor products of M copies of the noisy quantum state to compute an expectation value of the target observable with respect to an entangled quantum state, wherein the entangled quantum state is given by ρM/Tr(ρM) where ρ represents the noisy quantum state, M≥1, and eigenvalues corresponding to non-dominant eigenvectors of the noisy quantum state in the spectral decomposition of the entangled quantum state are suppressed exponentially in M; performing, by the quantum computer, measurements on tensor products of M copies of the noisy quantum state to compute an expectation value of an identity operator with respect to the entangled quantum state; and dividing, by the classical processor, the computed expectation value of the target observable with respect to the entangled quantum state by the expectation value of the identity operator with respect to the entangled quantum state to determine the error-mitigated expectation value of the target observable with respect to the noisy quantum state, wherein the error-mitigated expectation value of the target observable with respect to the noisy quantum state comprises a reconstructed expectation value of the target observable with respect to a purified version of the noisy quantum state and approximates an expectation value of the target observable computed by the quantum computer in an absence of the stochastic errors and noise on the quantum computer.

2. The method of claim 1, wherein noise experienced by each copy of the noisy quantum state comprises a same form and strength.

3. The method of claim 1, wherein performing measurements on tensor products of M copies of the noisy quantum state to compute expectation values of the target observable with respect to an entangled quantum state comprises: for each of a first number of measurement repetitions, measuring a product of i) the target observable and ii) a cyclic shift operator with respect to a tensor product of M copies of the noisy quantum state to obtain a first measurement outcome for each repetition; computing, using the first measurement outcomes obtained from the first number of measurement repetitions, an expectation value of the product of i) the target observable and ii) a cyclic shift operator with respect to the tensor product of M copies of the noisy quantum state; and normalizing the expectation value of the product of i) the target observable and ii) a cyclic shift operator with respect to the tensor product of M copies of the noisy quantum state to obtain the error-mitigated expectation value of the target observable with respect to the noisy quantum state.

4. The method of claim 3, wherein normalizing the expectation value of the target observable comprises: for each of a second number of measurement repetitions, measuring the cyclic shift operator with respect to a tensor product of M copies of the noisy quantum state to obtain a second measurement outcome for each repetition; computing, using the second measurement outcomes obtained from the second number of measurement repetitions, an expectation value of the cyclic shift operator with respect to a tensor product of M copies of the noisy quantum state; and dividing the expectation value of the product of i) the target observable and ii) a cyclic shift operator with respect to the tensor product of M copies of the noisy quantum state by the expectation value of the cyclic shift operator with respect to the tensor product of M copies of the noisy quantum state.

5. The method of claim 1, wherein performing measurements on tensor products of M copies of the noisy quantum state comprises performing parallel measurements.

6. The method of claim 1, wherein M≥2 and the target observable acts on a single qubit.

7. The method of claim 6, wherein performing measurements on tensor products of M copies of the noisy quantum state comprises, for each of multiple measurement repetitions, wherein the multiple measurement repetitions comprise K measurement repetitions: applying a diagonalization operator to a tensor product of M copies of the noisy quantum state to obtain an evolved quantum state, wherein the diagonalization operator diagonalizes i) a cyclic shift operator and ii) a product of a symmetrized version of the target observable and the cyclic shift operator, and measuring a product of the target observable and the cyclic shift operator with respect to the evolved quantum state to obtain a respective measurement outcome for the repetition for each qubit in the evolved quantum state.

8. The method of claim 7, further comprising: computing an expectation value of the product of the target observable and the cyclic shift operator with respect to the evolved quantum state using the measurement outcomes obtained from the K measurement repetitions; computing an expectation value of the cyclic shift operator with respect to the evolved quantum state using the measurement outcomes obtained from the K measurement repetitions; and dividing the expectation value of the product of the target observable and the cyclic shift operator with respect to the evolved quantum state by the expectation value of the cyclic shift operator with respect to the evolved quantum state.

9. The method of claim 7, wherein application of the cyclic shift operator couples qubits in a first copy of the noisy quantum state to corresponding qubits in each other copy of the noisy quantum state.

10. The method of claim 1, wherein performing measurements on tensor products of M copies of the noisy quantum state comprises performing serial measurements.

11. The method of claim 1, wherein M≥2 and the target observable acts on two or more qubits, the target observable comprising multiple tensor products of one-qubit operators.

12. The method of claim 11, wherein performing measurements on tensor products of M copies of the noisy quantum state comprises: for each tensor product of one-qubit operators: for each of a first number of measurement repetitions: applying a first diagonalization operator to a tensor product of the M copies of the noisy quantum state to obtain an evolved quantum state, wherein the first diagonalization operator diagonalizes a product of i) the tensor product of one-qubit operators and ii) the cyclic shift operator, and measuring a product of the tensor product of one-qubit operators and the cyclic shift operator with respect to the evolved quantum state to obtain a respective first measurement outcome for the repetition of the first number of repetitions for each qubit in the evolved quantum state; for each of a second number of measurement repetitions: applying a second diagonalization operator to a tensor product of the M copies of the noisy quantum state to obtain a second evolved quantum state, wherein the second diagonalization operator diagonalizes the cyclic shift operator, and measuring the cyclic shift operator with respect to the second evolved quantum state to obtain a respective second measurement outcome for the repetition of the second number of repetitions for each qubit in the second evolved quantum state.

13. The method of claim 12, further comprising: computing an expectation value of a product of the target observable and the cyclic shift operator using the first measurement outcomes obtained from the first number of measurement repetitions, computing an expectation value of the cyclic shift operator using the second measurement outcomes obtained from the second number of measurement repetitions; and dividing the expectation value of the product of the target observable and the cyclic shift operator by the expectation value of the cyclic shift operator.

14. The method of claim 1, wherein performing measurements on tensor products of M copies of the noisy quantum state to compute an expectation value of the target observable with respect to an entangled quantum state comprise performing ancilla-assisted measurements.

15. The method of claim 14, wherein the target observable comprises a single qubit operator or a multi-qubit operator.

16. The method of claim 14, wherein M≥2 and performing ancilla assisted measurements comprises, for each of multiple measurement repetitions: preparing two quantum registers, where each quantum register comprises the noisy quantum state and an ancilla qubit prepared in the 0 state; performing a non-destructive measurement of a cyclic shift operator, comprising: applying a Hadamard gate to the ancilla qubit, applying a cyclic shift operator to the two quantum registers conditioned on the ancilla qubit being in the 1 state, and measuring the ancilla qubit in the X basis to obtain a first respective measurement outcome; and measuring qubits in the two quantum registers with respect to a symmetrized version of the target observable to obtain respective second measurement outcomes.

17. The method of claim 16, further comprising: computing an expectation value of the target observable using the second measurement outcomes; and computing an expectation value of the cyclic shift operator using the first measurement outcomes; and dividing the expectation value of the target observable by the expectation value of the cyclic shift operator.

18. An apparatus comprising: one or more classical processors; and one or more quantum computing devices in data communication with the one or more classical processors, wherein the quantum computing hardware comprises: one or more qubit registers, each qubit register comprising one or more qubits, and a plurality of control devices configured to operate the one or more qubit registers; wherein the apparatus is configured to perform the method of determining an error-mitigated expectation value of a target observable with respect to a noisy quantum state, the method comprising: obtaining, by the one or more quantum computing devices, multiple copies of the noisy quantum state, wherein the noisy quantum state comprises stochastic errors resulting from noise on the quantum computer; performing, by the one or more quantum computing devices, measurements on tensor products of M copies of the noisy quantum state to compute an expectation value of the target observable with respect to an entangled quantum state, wherein the entangled quantum state is given by ρM/Tr(ρM), where ρ represents the noisy quantum state, M≥1, and eigenvalues corresponding to non-dominant eigenvectors of the noisy quantum state in the spectral decomposition of the entangled quantum state are suppressed exponentially in M; performing, by the one or more quantum computing devices, measurements on tensor products of M copies of the noisy quantum state to compute an expectation value of an identity operator with respect to the entangled quantum state; and dividing, by the one or more classical processors, the computed expectation value of the target observable with respect to the entangled quantum state by the expectation value of the identity operator with respect to the entangled quantum state to determine the error-mitigated expectation value of the target observable with respect to the noisy quantum state, wherein the error-mitigated expectation value of the target observable with respect to the noisy quantum state comprises a reconstructed expectation value of the target observable with respect to a purified version of the noisy quantum state and approximates an expectation value of the target observable computed by the quantum computer in an absence of the stochastic errors and noise on the quantum computer.